Wigan chairman Dave Whelan claims manager Roberto Martinez has been made an offer by managerless Liverpool but will await a further meeting before making his decision.
The Latics boss met with Liverpool's principal owner John Henry in Miami yesterday to talk about the vacant Anfield job after interrupting a family holiday in the Caribbean.
It was expected Fenway Sports Group would speak to other candidates on a short-list in the coming days but Whelan suggested developments had progressed.
He did, however, say Martinez - who would be expected to work under a technical director - wanted to negotiate a greater degree of control of footballing issues.
"I think what he has said is, 'Thank you for your offer but I only work under the following rules: I have to have complete control of football'," Whelan told Sky Sports News.
"That's normally the way in England, that's how we all work.
"I think they're considering his request and Roberto's considering their offer and they're going to talk again on Tuesday.
"Hopefully he stays with us but wherever he goes, he's 100% dedicated to that football club."
Liverpool have declined to comment on Whelan's claims.
It was anticipated the process of finding a replacement for Kenny Dalglish would have picked up pace next week with Henry and chairman Tom Werner likely to travel to England.
Martinez remains the favourite with the bookmakers, although that could have something to do with the fact he is the only contender publicly confirmed as being in the running.
Former Porto and Chelsea boss Andre Villas-Boas appears to have dropped out of the reckoning but it has been reported the Reds have not ruled out a second approach for Swansea's Brendan Rodgers after he declined their initial invitation to open preliminary talks.
Whelan, who set his manager a deadline of June 5 to decide his future, said yesterday he was not convinced Martinez would take the Liverpool job because of the structure under which he would be expected to work.
Former Wigan and Liverpool goalkeeper Chris Kirkland believes the Spaniard would regret not seizing the opportunity should it arise.
"What he'll be thinking to himself is: 'Will I get this opportunity again?"' said the 31-year-old, who had three seasons under Martinez at Wigan before agreeing a move to newly-promoted npower Championship side Sheffield Wednesday this week.
Kirkland added in the Wigan Evening Post: "The Liverpool job is still one of the biggest in the world - and it doesn't get offered to you every day.
"It's a massive job and it will be a massive opportunity for somebody.
"Whether that will be Roberto remains to be seem.
"But my gut feeling is that he'll go - there's been too much speculation for far too long for something not to be happening."
Martinez has failed to ignite the optimism of Liverpool fans, many of whom believe a more experienced manager should be tasked with getting the club back into the Champions League.
Former striker Peter Beardsley hopes the Spaniard, if appointed, will be able to win them over with his style of football.
However, the former England international believes FSG rushed into the decision to sack Dalglish after the club's eighth-placed finish this season.
"I think they panicked by getting rid of Dalglish," Beardsley told talkSPORT.
"I think they should have given him another year and seen where it goes.
"But Roberto has done well at Swansea and Wigan, he's come through the system and his type of football will appeal to the Liverpool fans."